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Perfect and Related Codes
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OUTLINE [1] Some bounds for codes [2] Perfect codes [3] Hamming codes
[4] Extended codes [5] The extended Golay code [6] Decoding the extended Golay code [7] The Golay code [8] Reed-Muller codes [9] Fast decoding RM(1,m)
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Perfect and Related Codes
[1] Some bounds for codes 1. The number of word of length n , weight t 2. Theorem 3.1.1
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Perfect and Related Codes
3. Theorem Hamming bound(upper bound) C: a code of length n, distance d = 2t+1 or 2t+2 Eg Give an upper bound of the size of a linear code C of length n=6 and distance d=3 So but the size of a linear code C must be a power of 2 so
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Perfect and Related Codes
4. Theorem Singleton bound(upper bound) For any (n, k, d) linear codes, d-1≦ n-k (i.e. k ≦ n-d+1 or |C| ≦2n-d+1 ) <pf> the parity check matrix H of an (n,k,d) linear code is an n by n-k matrix such that every d-1 rows of H are independent. Since the rows have length n-k, we can never have more than n-k independent row vectors. Hence d-1≦ n-k. 5. Theorem 3.1.8 For a (n, k, d) linear code C, the following are equivalent: d = n-k+1 Every n-k rows of parity check matrix are linearly independent Every k columns of the generator matrix are linearly independent C is Maximum Distance Separable(MDS) (definition: if d=n-k+1)
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Perfect and Related Codes
6. Theorem Gilbert-Varshamov condition There exists a linear code of length n, dimension k and distance d if (<pf> design a parity check matrix under this condition. See Ex3.1.22) 7. Corollary Gilbert-Varshamov bound(lower bound) If n≠1 and d ≠1, there exists a linear code C of length n and distance at least d with (<pf> choose k such that then |C| = 2k = )
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Perfect and Related Codes
Eg Does there exist a linear code of length n=9, dimension k=2, and distance d=5? Yes, because Eg What is a lower and an upper bound on the size or the dimension, k, of a linear code with n=9 and d=5? G-V lower bound: |C| ≧ but |C| is a power of 2 so |C| ≧ 4 Hamming upper bound: |C| ≦ but |C| is a power of 2 so |C| ≦ 8
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Perfect and Related Codes
Eg Does there exists a (15, 7, 5) linear code? Check G-V condition G-V condition does not hold, so G-V bound does not tell us Whether or not such a code exists. But actually such a code does exist. (See BCH code later)
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Perfect and Related Codes
[2]. Perfect Codes 1. Definition: A code C of length n and odd distance d = 2t+1 is called perfect code if 2. Theorem 3.2.8 If C is perfect code of length code of length n and distance d = 2t+1, then C will correct all error pattern of weight less than or equal t
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Perfect and Related Codes
[3]. Hamming Codes 1. Definition: Hamming code of length 2r-1 A code of length n = 2r-1, r ≧2, having parity check matrix H whose rows of all nonzero vectors of length r Eg the Hamming code of length 7(r=3)
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Perfect and Related Codes
H contains r rows of weight one, so its r columns are linearly independent. Thus a Hamming code has dimension k=2r-1-r and contains 2k codewords. It is a perfect single error-correcting code (d=3) 1. Any two rows of H are lin. indep so d ≧3 2. 100…0, 010…0, and 110…0 are lin. dep. so d ≦3 and thus d=3 3. Attains the Hamming bound (t=1)
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Perfect and Related Codes
[4]Extended Codes 1. Definition: A code C* of length n+1 obtained from C of length n Construction: Let G: kxn and choose G*=[G, b] : kx(n+1) where the last col b of G* is appended so that each row of G* has even weight. Let H: nx(n-k) then H* = where j is the nx1 col of all ones. Why? where Gj+b=0
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Perfect and Related Codes
Eg 3.4.1 C: length 5 C*: length 6
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Perfect and Related Codes
[5]. The extended Golay code(C24) 1. Definition the linear code C24 with generator matrix G= [I, B] I: 12x12 identity matrix B:
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Perfect and Related Codes
2. Important facts of the extended Golary Code C24 Length n =24, dimension k= 12 , 212=4096 codewords Parity check matrix Another parity check matrix Another generator matrix [B, I] C24 self-dual; The distance of C is 8 C24 is a three-error-correcting code
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Perfect and Related Codes
[6]. Decoding the extended Golay code 1. Algorithm IMLD for Code C24 w: received word 1) s = wH 2) if wt(s)≦3 then u = [ s,0 ](error pattern) 3) if wt(s+bi) ≦2 for some row bi of B then u=[s+bi, ei] 4) compute the second syndrome sB 5) if wt(sB)≦3 then u=[0, sB] 6) if wt(sB+bi)≦2 for some row bi of B then u=[ei, sB+bi] 7) if u is not yet determined then request retransmission
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Perfect and Related Codes
2. Eg decode w = ,
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[7]. The Golay code 1. C23 Removing a digit from every word in C24 : 12 x 11 matrix by deleting the last column of B G: 12x23 matrix, G=[I12, ] Length n=23, dimension k=12, 212=4096 words, distance d=7 Perfect code, three-error-correcting code Algorithm (decoding) Form w0 or w1, which has odd weight Decode wi(I is 0 or 1) using Algorithm to a codeword c in C24 Removing the last digit from c
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Perfect and Related Codes
[8]. Reed-Muller codes 1. r-th order, length 2m, 0≦r≦m, RM(r, m) RM(0, m)={00…0, 11…1}, RM(m, m)= RM(r, m) = 2. Eg 3.8.1
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Perfect and Related Codes
3. Generator matrix G of RM(r, m) 4. Eg Find G(1,3)
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5. The properties of RM(r,m)
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[9]. Fast decoding for RM(1,m) 1. The Kronecker product of matrices A x B = [aijB] Eg 3.9.1 Definition:
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Perfect and Related Codes
2. Algorithm decoding the RM(1,m) code codeword v = presumed message m x G(1,m)
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Eg m=3, w= codeword
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